This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337991 #19 Nov 01 2024 13:56:49
%S A337991 1,1,1,1,2,1,2,5,4,1,4,13,15,7,1,9,35,52,36,11,1,21,96,175,160,75,16,
%T A337991 1,51,267,576,655,415,141,22,1,127,750,1869,2541,2030,952,245,29,1,
%U A337991 323,2123,6000,9492,9156,5488,1988,400,37,1,835,6046,19107,34476,38976,28476,13356,3852,621,46,1
%N A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)*C(n+m-i,i-1)*C(n+m-i,m)/n, with T(0,0)=1.
%H A337991 G. C. Greubel, <a href="/A337991/b337991.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A337991 G.f.: ( 1 - x*(y-1)- sqrt(x^2*(y^2-2*y-3) - 2*x*(y+1) + 1) )/(2*x).
%F A337991 From _G. C. Greubel_, Oct 31 2024: (Start)
%F A337991 T(n, k) = binomial(n, 1-k)*binomial(n+k-1, k)*Hypergeometric3F2([1-n, (1 -n -k)/2, (2-n-k)/2], [2-k, 1-n-k], 4), with T(0, 0) = 1.
%F A337991 T(n, 0) = A086246(n+1).
%F A337991 T(n, n-1) = A000124(n-1), n >= 1.
%F A337991 T(n, n-2) = A006008(n-1), n >= 2.
%F A337991 T(n, n-3) = (1/72)*(n^4 -6*n^3 +47*n^2 -114*n +144)*binomial(n-1,2), n >= 3.
%F A337991 T(n, n-4) = (1/480)*(n-2)*(n^4 -8*n^3 +99*n^2 -332*n +960)*binomial(n-1,3), n >= 4.
%F A337991 Sum_{k=0..n} T(n, k) = A025227(n+1).
%F A337991 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
%F A337991 Sum_{k=0..floor(n/2)} T(n-k, k) = A102407(n).
%F A337991 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A019590(n+1). (End)
%e A337991 Triangle begins as:
%e A337991 1;
%e A337991 1, 1;
%e A337991 1, 2, 1;
%e A337991 2, 5, 4, 1;
%e A337991 4, 13, 15, 7, 1;
%e A337991 9, 35, 52, 36, 11, 1;
%e A337991 21, 96, 175, 160, 75, 16, 1;
%e A337991 51, 267, 576, 655, 415, 141, 22, 1;
%e A337991 ...
%t A337991 T[0, 0] = 1; T[n_, m_] := Sum[Binomial[n, i - m] * Binomial[n + m - i, i - 1] * Binomial[n + m - i, m]/n, {i, 1, n}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Amiram Eldar_, Oct 06 2020 *)
%o A337991 (Maxima)
%o A337991 T(n,m):=if m=n then 1 else if n=0 then 0 else sum(binomial(n,i-m)*binomial(n+m-i,i-1)*binomial(n+m-i,m),i,1,n)/n;
%o A337991 (Magma)
%o A337991 B:=Binomial;
%o A337991 A337991:= func< n,k | n eq 0 select 1 else (1/n)*(&+[B(n, j-k)*B(n+k-j, j-1)*B(n+k-j, k): j in [1..n]]) >;
%o A337991 [A337991(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 31 2024
%o A337991 (Python)
%o A337991 def A337991(n,k):
%o A337991 b=binomial
%o A337991 if n==0: return 1
%o A337991 else: return (1/n)*sum(b(n, j-k)*b(n+k-j, j-1)*b(n+k-j, k) for j in range(1,n+1))
%o A337991 # SageMath
%o A337991 flatten([[A337991(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 31 2024
%Y A337991 Cf. A001006, A005773, A086246, A132894.
%Y A337991 Diagonals include: A000124, A006008.
%Y A337991 Sums include: A000007 (signed row), A019590 (signed diagonal), A025227 (row), A102407 (diagonal).
%K A337991 nonn,tabl
%O A337991 0,5
%A A337991 _Vladimir Kruchinin_, Oct 06 2020